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E130 



THE EFFECT OF PROMOTION RATES 
ON SCHOOL EFFICIENCY 



BY 



LEONARD P. AYRES, PH.D. 



R-S 
•F- 



Reprinted from the 
American School Board Journal, May, 19 13 

Division of Education 
Russell Sage Foundation 

400 Metropolitan Tower, New York City 



Price 5 Cents 



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The Effect of Promotion Rates on School 
Efficiency 

Leonard P. Ayres, Ph.D., 

Director, Division of Education, Russell Sage Foundation 

Five years ago 84 out of each 100 children in the elementary 
grades of 16 of our large cities were promoted at the end of the 
school term. Last year 88 out of each 100 children in the same 
16 cities were promoted at the end of the school year. There is 
abundant evidence that a similar general increase in promotion 
rates is taking place throughout the country. Since this affects 
the school lives of thousands of our children, school officials are 
confronted by the problem of finding out in what ways and to 
what degree it affects the children and the schools. 

If we were considering problems of business or transportation, 
the general principles involved would seem quite simple. If a 
train making regular trips between two far distant cities runs at 
less than its normal rate of speed, it will take more than the nor- 
mal amount of time to cover the distance. Similarly in a school 
system the average per cent of promotion can be turned into 
terms of time so as to tell us how long it will take the average 
child to complete the eight grades of the elementary course if he 
stays to do it. 

Time Required to Complete Eight Grades 
If every child in a school system should complete a full grade 
and be promoted at the end of every school year, the impossible 
ideal or normal rate of 100 per cent of promotions would be at- 
tained and the average child would complete eight grades in eight 
years. If only 80 per cent of the children are promoted at the 
end of each year, the average rate of progress is f of the 
normal and hence the average child will require -f of eight years 
or ten years to complete the course. This computation, like 
all the following ones, is based on the assumptions that the pro- 
motion rate applies equally to the children throughout the 
course, that none die, and none leave before finishing. 

3 



If we can compute by the simple process indicated that a 
promotion rate of 80 per cent applicable to all the children means 
that the average child will take 10 years to complete eight grades, 
we can figure by the same method what other promotion rates 
mean in terms of the time required to complete the course. The 
results are presented in column A of the table and section A of 
the diagram. 

The Size of the School Plant 
Suppose that conditions require our railroad train to trans- 
port a given amount of merchandise each year. It is clear that 
in proportion as the speed of the trips is reduced extra cars will 
have to be added to carry the load. 

The same principle applies to our school systems. The com- 
munity hands over to the schools a new crop of children each year 
and they all have to be accommodated. In proportion as their 
rate of progress through the grades is reduced, the size of the 
school plant, the amount of equipment, and the number of teach- 
ers must be increased. If the average rate of promotions is 80 
per cent, or f of the normal, the size of the school plant 
required to accommodate the children must be f or 125 per 
cent of the theoretic normal. When this is translated into 
terms of dollars, the results for even a small city are impressive. 
Figures showing the effect of different promotion rates on the 
size of the school plant are presented in column B of the table. 
The same data are shown in graphic form in section B of the 
diagram. 

Number of Failures per 1,000 Children 
When we say that the promotion rate is 90 per cent, we mean 
that of each 1,000 children 900 are promoted and 100 fail of 
promotion. In a system having annual promotions there are 
seven promotion times before graduation in the eight years of 
elementary schooling. If during this entire period the average 
promotion rate were always 90 per cent, 100 out of the 1,000 
children would fail at each of the seven promotion periods and 
there would be 700 failures among the 1,000 children in the eight 
years. 

In a similar way the corresponding number of failures may be 
computed for each promotion percentage, because each time we 
drop the rate by one per cent, there will be 70 more failures among 



the 1 ,000 children during the eight years. Following this method 
the number of failures among each thousand children in eight 
years has been computed for each promotion percentage from 
ioo down to 60. The results are presented in figures in column 
C of the table and illustrated in section C of the diagram. These 
data are an index of the degree to which the children are being 
trained in the habit of failure. 

Promotions and Repeaters 

The child who is not promoted does the work of the grade 
over again. He is a repeater. But the number of children in 
each thousand who repeat grades during eight years of school 
life is not the same as the number of repetitions, for one child may 
repeat grades several times. The number of children who will 
repeat grades one or more times during eight years at any given 
constant promotion rate may be computed by constructing a 
grade distribution for a hypothetical school system in which 1 ,000 
children enter and are steadily promoted annually at, say, an 
80 per cent rate, none die and none drop out. 

Thus the first year there will be 1,000 children in the first 
grade. Then 80 per cent of them are promoted and the second 
year there will be 200 of them left to repeat the first grade and 
800 will go on to the second grade. In the third year the promo- 
tion rate of 80 per cent again applies to all of them and as a result 
40 are left in the first grade, 320 are in the second grade, and 
only 640 have gone on to the third grade. 

By repeating these computations we can carry our thousand 
children through eight years of elementary schooling and find 
out how many have not repeated grades and how many have. 
By the same method we may ascertain the corresponding figures 
for each promotion rate. The data showing the number of 
children in each thousand failing and repeating grades in the 
course of eight years of elementary schooling at each promotion 
rate from 100 per cent down to 60 per cent are shown in column 
D of the table and section D of the diagram. These figures are 
an index of the number of children who are trained in the habit 
of failure. 

Promotions and Over-Age Children 
All children who are above the normal ages for their grades 
are over-age for one or both of two reasons : either they entered 



6 

school late or they made slow progress. If we eliminate the first 
cause by assuming that all the children begin school at the normal 
age of from six and one-half to seven years, we may compute for 
each promotion percentage the number of children in each thou- 
sand who will be over-age at the end of eight years. Since low 
promotion rates mean slow average progress, one of the results 
will be the production of over-age children just as inevitably as 
a slow speed for a train carrying perishable merchandise will 
result in an increased proportion of damaged goods. 

By means of hypothetical age and grade tables, the percent- 
age of over-age children resulting from each promotion per- 
centage from ioo to 60 has been computed. In every case these 
computations are based on the propositions that 1,000 children 
enter school each year at the age of six and one-half to seven 
years, the promotion rate is constant for eight years, no children 
die and none drop out, and the count of over-age pupils is made 
on the last day of the year before promotion. The results are 
presented in column E of the table and section E of the diagram. 

Annual and Semi-Annual Promotions 
School administrators often speculate as to the results of 
changing from a system of annual promotions to a semi-annual 
one. From a purely mathematical point of view the answer is 
that the change has no effect whatever on the average progress 
rate of the children. If, for example, 80 per cent of all the chil- 
dren are promoted every time, it makes no difference in the aver- 
age progress of the whole group whether the promotions take 
place once a year or ten times a year. But while the average prog- 
ress of the children is not changed, their distribution through 
the different grades is materially affected. 

If 100 children are promoted annually at a steady 80 per cent 
rate for eight years, we shall find them at the end of that time 
distributed through the grades as follows: 

Grade Children 

4 3 

5- 11 

6 27 

7 38 

8 21 

Total 100 

If the promotions take place semi-annually instead of an- 
nually, the distribution of these 100 children at the end of the 
eight years will be as follows: 



7 

Grade Children 

5B 3 

6A 10 

6B 19 

7A 26 

7B 24 

8A 14 

8B _\ 

Total 100 

The difference between the two distributions is marked. 
Under the annual system 21 children have completed the eighth 
grade while, under the semi-annual system, only four have com- 
pleted it. On the other hand the annual system has left three chil- 
dren in the fourth grade while the semi-annual one has left none 
lower than the 5B grade. The annual system has carried more 
children through on schedule time but left more far back in the 
grades, while the semi-annual system has carried fewer all the 
way through but left fewer badly retarded. The annual system 
has bettered conditions for the few while the semi-annual one has 
bettered them for the many. The difference between the two 
results may be more clearly seen if we picture the resulting dis- 
tributions by representing each child in the fourth grade by a 
figure "4," each one in the fifth grade by a figure "5" and so on 
for those in the other grades. This gives us a distribution under 
the annual promotion system as follows: 

DISTRIBUTION BY GRADES OF 100 CHILDREN AFTER EIGHT 
YEARS OF ANNUAL PROMOTIONS AT A CONSTANT RATE OF 80 
PER CENT. EACH DIGIT REPRESENTS ONE CHILD AND ITS 
DENOMINATION SHOWS THE GRADE THE CHILD IS IN 

77777777 
6666666 7777777777 8 

5 6666666666 7777777777 8888888888 

444 5555555555 6666666666 7777777777 8888888888 

(three) (eleven) (twenty-seven) (thirty-eight) (twenty-one) 

In a similar way the results of the semi-annual system at the 
same rate may be represented as follows: 

DISTRIBUTION BY GRADES OF 100 CHILDREN AFTER EIGHT 
YEARS OF SEMI-ANNUAL PROMOTIONS AT A CONSTANT RATE 
OF 80 PER CENT. EACH DIGIT REPRESENTS ONE CHILD AND 
ITS DENOMINATION SHOWS THE GRADE THE CHILD IS IN 

7 
77777 7777 
6666 77777 77777 
66666 77777 77777 8888 
66666 66666 77777 77777 88888 
555 66666 66666 77777 77777 88888 8888 
BAB A B A B 

(three) (ten) (nine- (twenty- (twenty- (four- (four) 

teen) six) four) teen) 



EFFECTS OF DIFFERENT ANNUAL PROMOTION RATES IN A 
SCHOOL SYSTEM IN WHICH 1,000 CHILDREN ENTER EACH 
YEAR, NONE DIE, AND NONE DROP OUT 





A 


B 


c 


D 


E 


Promo- 
tion rate 


Years re- 
quired for 
average 
child to 


Per cent of 

normal size 

required 


Failures 
among each 
1,000 children 


Children in 
each 1,000 
failing in 


Per cent of 

children 

above 




complete 8 
grades 


for school 
plant 


in eight 
years 


eight years 


normal age 
for grades 


IOO 


8.00 


IOO.O 











99 


8.08 


IOI.O 


70 


68 


34 


98 


8.16 


I02.0 


I40 


132 


6-7 


97 


8.24 


IO3.O 


210 


192 


9.9 


96 


8-33 


104. 1 


280 


249 


12.9 


95 


8.42 


IO5.2 


350 


302 


15.9 


94 


8.50 


IO6.3 


420 


352 


18.7 


93 


8.60 


107.5 


490 


398 


21.4 


92 


8.69 


108.6 


560 


442 


24.O 


91 


8.78 


IO9.8 


63O 


483 


26.4 


90 


8.89 


III. I 


700 


522 


28.8 


89 


8.98 


II2.3 


770 


558 


31. 1 


88 


9.09 


II3-6 


84O 


591 


33-3 


87 


9.19 


1 14.9 


910 


623 


354 


86 


9.30 


II6.2 


980 


652 


374 


85 


9.41 


II7.6 


1050 


679 


39-4 


84 


9-52 


1 19.O 


1 120 


705 


4I.2 


83 


9-63 


120.4 


1 190 


729 


43-0 


82 


975 


I2i;9 


1260 


751 


44-8 


81 


9.87 


123.4 


1330 


771 


46.4 


80 


10.00 


I25.O 


1400 


790 


48.O 


79 


10.12 


126.5 


I470 


808 


49-5 


78 


10.27 


128.2 


I540 


824 


51.0 


77 


10.38 


129.8 


l6lO 


840 


52.4 


76 


10.52 


I3I-5 


1680 


854 


53-7 


75 


10.66 


133-3 


1750 


867 


55-0 


74 


10.81 


I35.I 


1820 


878 


56.2 


73 


10.95 


136.9 


I89O 


890 


574 


72 


11. 10 


138.8 


i960 


900 


58.6 


7i 


11.26 


I4O.8 


2030 


909 


59-7 


70 


11.42 


I42.8 


2100 


918 


60.7 


69 


11.59 


144.9 


2170 


926 


61.7 


68 


11.76 


I47.O 


224O 


933 


62.7 


67 


11.94 


149.2 


2310 


939 


63.6 


66 


12.12 


I5I-5 


2380 


945 


64.5 


65 


12.30 


153.8 


2450 


951 


654 


64 


12.50 


156.2 


2520 


956 


66.2 


63 


12.70 


158.7 


259O 


961 


67.0 


62 


12.90 


l6l. 2 


2660 


965 


67.8 


61 


13.11 


I63.9 


2730 


969 


68.6 


60 


13.33 


166.6 


2800 


972 


69.3 



A. Years required for 
average child to complete 
eight grades at each pro- 
motion rate from ioo to 
60 per cent. 



B. Per cent, of normal 
size required for school 
plant at each promotion 
rate from 100 down to 60 
per cent. 



C. Failures among each 
1,000 children in eight 
years at each promotion 
rate from 100 down to 60 
per cent. 



D. Children in each 1,000 
failing in eight years at 
each promotion rate from 
100 down to 60 per cent. 



E. Per cent, of children 
above normal age for 
grades at each promotion 
rate from 100 down to 60 
per cent. 



PAOMOT/OA/ RATES 

/CO 9S 96 €S 90 7S TO 6S 60' 


13 
12. 
II 
10 

9 




















































9 


















I 


















S 






















3000 
2.S00 

zooo 

ISOD 
tooo 

soo 








































































































































1000 

800 

600 
400 
LOO 





































































































70 
60 
SO 

90 

to 

10 





































































































































10 

The two figure-diagrams illustrate clearly the characteristic 
differences between the results of the two systems. Under the 
annual system more children have completed the course but more 
are seriously retarded. Under the semi-annual system fewer 
have completed the course but fewer are seriously retarded. 

By employing the methods outlined in the preceding sections, 
other comparisons between the effects of annual and semi-annual 
promotions may be made. These comparisons show that when 
all other factors are equal, semi-annual promotions result in lower 
percentages of over-age children than do annual promotions. 
Under semi-annual promotions more children fail but each fail- 
ure is less serious than under the annual system. 

Validity and Application of the Computations 
Six sets of computations have been presented. Five of them 
are designed to show the effects of each promotion rate from ioo 
per cent down to 60 per cent in 

A. Years required for average child to complete eight grades 

B. Per cent of normal size required for school plant 

C. Failures among each 1,000 children before completing 

eight grades 

D. Children in each 1,000 failing before completing eight 

grades 

E. Per cent of children above normal ages for grades. 
There are several reasons why the results are not absolutely 

valid. In each case one hypothesis on which the computations 
are based is that none of the children leave the system through 
death or elimination. Now in point of fact some children die in 
every school system and some leave without completing the 
course. The second factor is far more important than the first 
for relatively few die while very many drop out. In general 
children in city schools remain in the elementary grades about 
eight years. If at the end of that time they have nearly or quite 
completed the course, they are apt to continue to the high school. 
If they have not, they are apt to leave school never to return. 

For this reason the figures presented in columns A, B, and E 
are fairly trustworthy only when applied to conditions in the six 
lower grades. They do not apply so truly to conditions in the 
two upper grades for those grades are largely made up of the sur- 
vivors who remain after large numbers of the slower pupils drop 
out. 



II 

For practical purposes the comparative studies of annual 
and semi-annual promotions are probably the least valuable of 
those presented. In actual application the semi-annual pro- 
motion system not only has the advantage shown here of carry- 
ing the great bulk of the children forward more consistently, but 
the further one of giving the brighter children more opportunity 
to complete the course on schedule time, while at the same time 
penalizing the slower ones less severely. 

What a Change of One Per Cent Means 
The importance of small changes in promotion rates may be 
best illustrated by figuring the results of a change of one per cent, 
say for example from 80 per cent to 81 per cent, in the promotion 
rate in the elementary schools of a small city. Let us suppose 
that 1,000 children enter the elementary schools each year, the 
annual per capita cost for schooling is $40, and the buildings, 
grounds, and equipment have a value of $200 per child. 

Under these conditions, the change in the promotion rate from 
80 per cent to 81 per cent will have the following results: The 
time saved by each 1 ,000 children if they complete the elementary 
course will amount to 130 years of schooling, which means a 
saving of $5,200 annually. The plant required to accommodate 
the children will be decreased by about $25,600 worth, and the 
salaries. of four teachers will be saved. The number of failures 
among the 1,000 children during eight years of school life will be 
reduced by 70, while the number of children failing during that 
period will be lessened by 19. The number of over-age children 
in the grades will be reduced by 220. These figures strikingly 
illustrate the importance of even the smallest changes in promo- 
tion rates. 

Summary 

1. Small differences in promotion rates have seriously im- 
portant results, both human and material. 

2. As the promotion rate falls, the time required for the aver- 
age child to complete the course increases. 

3. The lower the promotion rate, the larger must the school 
plant be to accommodate the children. 

4. For each per cent that the promotion rate falls, there are 
70 more failures among each 1,000 children in eight years of 
schooling. 



12 

5. As the promotion rate falls, the number of children failing 
in the course of eight years rapidly increases. 

6. The lower the promotion rate, the higher will be the per- 
centage of over-age children. 

7. The annual promotion system tends theoretically to carry 
more children through the course on schedule time than does the 
semi-annual system but it leaves more of them seriously retarded. 



The figures and diagrams that have been presented impress- 
ively illustrate the great importance of small differences in pro- 
motion rates. In its educational significance, a 75 per cent rate 
is an entirely different matter from one of 80 per cent and this 
again is not at all the same as one of 90 or 95 per cent. Every 
change of even one per cent either up or down is reflected by 
great and far reaching consequences in terms of dollars, teachers, 
plant, equipment and children's time. Most important of all, 
it has its very great effect on the degree to which the children are 
trained in habits of success and failure. 



13 

Some Publications on Measurements in Education 

Issued by the Division of Education, 

Russell Sage Foundation* 

No. 61. The Relation of Physical Defects to School 
Progress. 
A statistical study based on 6708 cases. 9 pp. 

No. 107. The Binet-Simon Measuring Scale for Intelli- 
gence: Some Criticisms and Suggestions. 
A critical study of these tests and suggestions as to 
their adaptation to our conditions. 11 pp. 

No. 108. The Identification of the Misfit Child. 

Data from a study of the age and progress records 
of school children in 29 cities. 1 1 pp. 

No. no. The Relative Responsibility of School and So- 
ciety for the Over-age Child. 
Data from a study of the age and progress records 
of school children in 29 cities. 6 pp. 

No. 112. The Relation Between Entering Age and Sub- 
sequent Progress Among School Children. 
Evidence from three investigations of the problem 
"What is the best age to send a child to school?" 
9PP- 
No. 113. A Scale for Measuring the Quality of Hand- 
writing of School Children. 
A quantitative study of legibility. (Report 5 
cents. Scale 5 cents.) 16 pp. 

No. 116. The Measurement of Educational Processes 
and Products. 
An account of the development of the quantitative 
method in education. 9 pp. 

No. 124. A Comparative Study of Public School Systems 
in the Forty-eight States. 
Fifteen tests of educational efficiency and accom- 
plishment. 32 pp. (Price 15 cents.) 

No. E 126. The Spelling Vocabularies of Personal and 
Business Letters. 
An analysis of 2000 letters. 16 pp. 

No. E 128. Psychological Tests in Vocational Guidance. 
Analysis of the problem, with brief sketch of the 
progress already made towards its solution. 6 pp. 

No. E 130. The Effect of Promotion Rates on School 
Efficiency. 
A method and tables for measuring differences in 
the effects of promotion rates. 12 pp. 
* A charge of five cents a copy is made for these publications unless other- 
wise specified. 



WAY ^2 1913 



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